Loop correction and resummation of vertex functions for a self interacting scalar field in the de Sitter spacetime

We consider a massless and minimally coupled self interacting quantum scalar field theory in the inflationary de Sitter background of dimension four. The self interaction potential is taken to be either quartic, $\lambda {\varphi }^4/4!$, or quartic plus cubic, $\lambda {\varphi }^4/4!+\beta {\varphi }^3/3!$ ($\lambda >0$). We compute the four and three point vertex functions up to two loop. The purely local or partly local part of these renormalised loop corrected vertex functions grow unboundedly after sufficient number of de Sitter $e$-foldings, due to the appearances of secular logarithms. We focus on the purely local part of the vertex functions and attempt a resummation of them in terms of the dynamically generated mass of the scalar field at late times. Such local logarithms have sub-leading powers compared to the non-local leading ones which can be resummed via the stochastic formalism. The variation of these vertex functions are investigated with respect to the tree level couplings numerically. Since neither the secular effect, nor the dynamical generation of field mass is possible in the Minkowski spacetime, the above phenomenon has no flat spacetime analogue. We have also compared our result with the ones that could be found via the recently proposed renormalisation group techniques. All these results suggest that at late times the value of the non-perturbative vertex function should be less than the tree level coupling.